Problem: All of the 3rd grade teachers and students from Gardner Bullis went on a field trip to an archaeology museum. Tickets were $$8.50$ each for teachers and $$4.50$ each for students, and the group paid $$39.50$ in total. The next month, the same group visited a science museum where the tickets cost $$25.50$ each for teachers and $$10.00$ each for students, and the group paid $$101.00$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+4.5y = 39.5}$ ${25.5x+10y = 101}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-25.5x-13.5y = -118.5}$ ${25.5x+10y = 101}$ Add the top and bottom equations together. $ -3.5y = -17.5 $ $ y = \dfrac{-17.5}{-3.5}$ ${y = 5}$ Now that you know ${y = 5}$ , plug it back into $ {8.5x+4.5y = 39.5}$ to find $x$ ${8.5x + 4.5}{(5)}{= 39.5}$ $8.5x+22.5 = 39.5$ $8.5x = 17$ $x = \dfrac{17}{8.5}$ ${x = 2}$ You can also plug ${y = 5}$ into $ {25.5x+10y = 101}$ and get the same answer for $x$ ${25.5x + 10}{(5)}{= 101}$ ${x = 2}$ There were $2$ teachers and $5$ students on the field trips.